Atom density matrix formalism

The polarization results from the dipole moments, t r,t), of the gas atoms induced by the electric field (7.7). It can be calculated in the framework of the atomic density matrix formalism. The main features of the phenomena, however, are also reproduced by the classical model of a damped oscillator. We are interested in the case where the frequency of light, m, is close to the atomic transition frequency, coq- The time evolution of the atomic dipole moment can be described, therefore, by the equation for an oscillator driven by the external force eEf (r, f) (Landau and Lifshitz 1978)... 

Separability can be exploited even with admission of relativistic effects, by using the standard density matrix formalism with a simple extension to admit 4-component Dirac spin-orbitals this opens up the possibility of performing ab initio calculations, with extensive d, on systems containing heavy atoms. 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

Since we are primarily interested in the conditional quantum state of the atomic ensemble, we can use the density-matrix formalism to calculate the fidelity with which we create the desired atomic state, Ins ) ... 

As described in Sec. 3.1, each Hartree-Fock iteration involves the construction of the Fock matrix for a given density matrix, followed by the diagonalization of the Fock matrix to generate a set of improved spin orbitals and thus an improved density matrix. Formally, the construction of the Fock matrix requires a number of operations proportional to K4, where K is the number of atoms (because the number of two-electron integrals scales as Al4). For large systems, however, this quartic scaling with K (i.e., with system size) can be reduced to linear by special techniques, as will now be discussed. 

By the same experimental technique, the temperature dependence of the nuclear spin relaxation rates was investigated for the radical cations of dimethoxy- and trimethoxybenzenes [89], The rates of these processes do not appear to be accessible by other methods. As was shown, l/Tfd of an aromatic proton in these radicals is proportional to the square of its hyperfine coupling constant. This result could be explained qualitatively by a simple MO model. Relaxation predominantly occurs by the dipolar interaction between the proton and the unpaired spin density in the pz orbital of the carbon atom the proton is attached to. Calculations on the basis of this model were performed with the density matrix formalism of MO theory and gave an agreement of experimental and predicted relaxation rates within a factor of 2. 

Within the density-matrix formalism (Vol. 1, Sect. 2.9) the coherent techniques measure the off-diagonal elements pab of the density matrix, called the coherences, while incoherent spectroscopy only yields information about the diagonal elements, representing the time-dependent population densities. The off-diagonal elements describe the atomic dipoles induced by the radiation field, which oscillate at the field frequency radiation sources with the field amplitude Ak(r, t). Under coherent excitation the dipoles oscillate with definite phase relations, and the phase-sensitive superposition of the radiation amplitudes Ak results in measurable interference phenomena (quantum beats, photon echoes, free induction decay, etc.). 

Note that an elegant theoretical way of describing observable quantities of a coherently or incoherently excited system of atoms and molecules is based on the density-matrix formalism. This formalism will not be described in detail here however, a basic summary is given in Box 2.6. 

Many experiments in laser spectroscopy depend on the coherence properties of the radiation and on the coherent excitation of atomic or molecular levels. Some basic ideas about temporal and spatial coherence of optical fields and the density-matrix formalism for the description of coherence in atoms are therefore discussed at the end of this chapter. 

The term decoherence describes the process by which the off-diagonal elements of the reduced density matrix tend to zero when evolving with time. Our objective is to reach an understanding of the molecular mechanisms governing decoherence with an atomic resolution. In addition we wish to be in a position to treat systems consisting of tens to thousands of atoms since the brute force simulation of the time evolution of p t) by the Liouville-von Neuman equation (p (i) = ih [H, p ]), the equivalent of the TDSE in the density matrix formalism, is out of question for such molecular systems. 

The HMO quantities q, and in (16.59) and (16.60) are closely related to the density matrix elements P [Eq. (13.165)] of SCF theory. We see that the density matrix elements with r = s (the diagonal elements) are equal to qr,. P = q. Althou HMO theory defines bond orders only for pairs of bonded atoms, if we formally define p by (16.60) for nonbonded pairs of atoms also, then the definitions yield = P -r P for real HMOs, p = Prs-... 

Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems. 

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